DV predicted by two IVs, vs. triangular pyramid

-- Diagram from Wiki

It is easier to imagine relation in three spatial vectors by their angles, than by their correlations. For standardized DV Y=\left(y_{1},y_{2},\dots,y_{N}\right)^{\tau} and IVs X_{1}=\left(x_{1,1},x_{2,1},\dots,x_{N,1}\right)^{\tau}, X_{2}=\left(x_{1,2},x_{2,2},\dots,x_{N,2}\right)^{\tau}, cosines of three angles of the triangular pyramid determinate the correlation matrix, thus, all statistics of the regressions Y=\beta_{1}X_{1}+\beta_{2}X_{2}+\varepsilon and Y=\beta_{1}X_{1}+\varepsilon . Unexpected but imaginative results on the impact of introducing X_{2} are --

1. Both IVs are nearly independent of DV. Togethor they predict DV almost perfectly (\angle YX_{1}=\angle YX_{2}=89^{\circ} and \angle X_{1}X_{2}=177.9^{\circ}).

2. Both IVs are almost perfectly correlated with DV. Togethor, one of the regressive coefficient is significantly negative (1^{\circ}, 0.6^{\circ} and 0.5^{\circ} respectively).

3. Redundancy (Cohen, Cohen, West, & Aiken, 2003) increases to full and then decreases to zero and even negative (\angle YX_{1}=60^{\circ}, \angle YX_{2}=45{}^{\circ} and \angle X_{1}X_{2} closes from 90^{\circ} to 45^{\circ} then to 15^{\circ}+\epsilon ).




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Cohen, J., Cohen, P., West, S. G., & Aiken, L. S.  (2003). Applied multiple regression/correlation analysis for the behavioral sciences(3rd ed.) Mahwah, NJ: Lawrence Erlbaum Associates.